文档介绍：1 Jean Berstel
2 Christophe Reutenauer
3 mutative Rational
4 Series With Applications
9 Pour Anne et Anissa
10 Preface
11 Formal power series have long been used in all branches of mathematics. They are
12 invaluable in algebra, analysis, combinatorics and in puter science.
13 Historically, the work of M.-P. Sch¨utzenberger in the algebraic theory of finite au-
14 tomata and the corresponding languages has led him to introduce mutative for-
15 mal power series. This appears in particular in his work with Chomsky on formal
16 grammars. This last point of view is at the origin of this book.
17 The first part of the book, composed of Chapters 1–4, is especially devoted to this
18 aspect: Formal power series may be viewed as formal languages with coefficients, and
19 finite automata (and more generally weighted automata) may be considered as linear
20 representations of the free monoid. In this sense, via formal power series, algebraic
21 theory of automata es a part of representation theory.
22 The first two chapters, contain general results and discuss in particular the equality
23 between rational and recognizable series (Theorem of Kleene–Sch¨utzenberger) and
24 the construction of the minimal linear representation. The exposition illustrates the
25 synthesis of linear algebra and syntactic methods inherited from automata theory.
26 The next two chapters are concerned with parison of some typical proper-
27 ties of rational (regular) languages, when they are transposed to rational series. First,
28 Chapters 3 describes the relationship with the family of regular languages studied in
29 puter science. Next, the chapter contains iteration properties for ratio-
30 nal series, also known as pumping lemmas, which are much more involved than those
31 for regular languages. Chapter 4 discusses rational expressions. It contains two main
32 results: the so-called “triviality” of rational identities over mutative ring and the
33 characterization of the